Today Let's Talk about Surface Brightness

Give the definition of surface brightness.

The surface brightness of an extended object, for example a galaxy, is the radiative flux per square arcseconds on the sky.


                                            ------- 
---------------------------- |   D   |
             d                            --------

D= area on the sky in [square arcseconds]
d= distance of the galaxy we're considering [parsecs]

This galaxy subtends an angle alpha on the sky:

$$\alpha =\frac {D}{d}$$

If I measure the luminosity L of all the stars into the area D I've the TOTAL flux:

$$F =\frac {L}{ 4 \cdot \pi \cdot (d)^2}$$

So I can define the surface brightness with this formula ( very important):

$$I = \frac{F}{(\alpha)^2} = \frac{F}{ \big(\frac{D}{d}\big) ^2}=\frac {L}{ (4 \pi \cdot (d)^{2}) \cdot \big(\frac{D}{d}\big)^{2}}=\frac {L}{ 4 \cdot \pi \cdot (D)^2}$$

$$I \bigg[\frac{mag} {arcseconds^2}\bigg]$$

References:
 1)http://asd.gsfc.nasa.gov/David.Davis//courses/phys315/week6/week6.pdf
 2)http://www.astro.spbu.ru/staff/resh/Lectures/lec1.pdf